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Theorem imaeq2 4845
Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)
Assertion
Ref Expression
imaeq2  |-  ( A  =  B  ->  ( C " A )  =  ( C " B
) )

Proof of Theorem imaeq2
StepHypRef Expression
1 reseq2 4782 . . 3  |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )
21rneqd 4736 . 2  |-  ( A  =  B  ->  ran  ( C  |`  A )  =  ran  ( C  |`  B ) )
3 df-ima 4520 . 2  |-  ( C
" A )  =  ran  ( C  |`  A )
4 df-ima 4520 . 2  |-  ( C
" B )  =  ran  ( C  |`  B )
52, 3, 43eqtr4g 2173 1  |-  ( A  =  B  ->  ( C " A )  =  ( C " B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1314   ran crn 4508    |` cres 4509   "cima 4510
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520
This theorem is referenced by:  imaeq2i  4847  imaeq2d  4849  ssimaex  5448  ssimaexg  5449  isoselem  5687  f1opw2  5942  fopwdom  6696  ssenen  6711  fiintim  6783  fidcenumlemrk  6808  fidcenumlemr  6809  sbthlem2  6812  isbth  6821  ennnfonelemp1  11825  ennnfonelemnn0  11841  ctinfomlemom  11846  ctinfom  11847  tgcn  12283  iscnp4  12293  cnpnei  12294  cnima  12295  cnconst2  12308  cnrest2  12311  cnptoprest  12314  txcnp  12346  txcnmpt  12348  metcnp3  12586
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